| We investigate some homological properties of graded algebras. If A is an R-algebra, then E( A) := ExtA(R, R) is an R-algebra under the cup product and is called the Yoneda algebra. (In most cases, we assume R is a field.) A well-known and widely-studied condition on E(A) is the Koszul property. We study a class of deformations of Koszul algebras that arises from the study of equivariant cohomology and algebraic groups and show that under certain circumstances these deformations are Poincare-Birkhoff-Witt deformations.;Some of our results involve the K2 property, recently introduced by Cassidy and Shelton, which is a generalization of the Koszul property. While a Koszul algebra must be quadratic, a K2 algebra may have its ideal of relations generated in different degrees. We study the structure of the Yoneda algebra corresponding to a monomial K2 algebra and provide an example of a monomial K2 algebra whose Yoneda algebra is not also K2 . This example illustrates the difficulty of finding a K2 analogue of the classical theory of Koszul duality.;It is well-known that Poincare-Birkhoff-Witt algebras are Koszul. We find a K2 analogue of this theory. If V is a finite-dimensional vector space with an ordered basis, and A := T (V)/I is a connected-graded algebra, we can place a filtration F on A as well as E(A). We show there is a bigraded algebra embedding Lambda : grF E( A) ↪ E(grF A). If I has a Grobner basis meeting certain conditions and grF A is K2 , then Lambda can be used to show that A is also K2 .;This dissertation contains both previously published and co-authored materials. |
